Bergman norm estimates of Poisson integrals

Abstract

On the half space ${\bf R}^n\times {\bf R}_+\,$, it has been known that harmonic Bergman space $b^p$ can contain a positive function only if $p>1+\frac 1n$. Thus, for $1\le p\le 1+\frac 1n$, Poisson integrals can be $b^p$-functions only by means of their boundary cancellation properties. In this paper, we describe what those cancellation properties explicitly are. Also, given such cancellation properties, we obtain weighted norm inequalities for Poisson integrals. As a consequence, under weighted integrability condition given by our weighted norm inequalities, we show that our cancellation properties are equivalent to the $b^p$-containment of Poisson integrals for $p$ under consideration. Our results are sharp in the sense that orders of our weights cannot be improved.